📚 Year 10 AQA Philosophy: Formula & Theorem Quick Reference Handbook | Year 10 AQA 哲学：公式定理速查手册
Philosophy has no equations in the way mathematics does, but it is built on powerful logical forms, argument schemas, and conceptual 'formulae' that you can apply again and again in exams. This handbook collects the most important of these – from formal logic rules to the core structures of ontological, design, utilitarian, and Kantian arguments. Memorising them will give you a reliable toolkit for analysis and evaluation questions.
哲学不像数学那样充满方程，但它建立在强有力的逻辑形式、论证框架和概念“公式”之上，你可以在考试中反复运用。这本手册汇集了其中最重要的内容——从形式逻辑规则到本体论、设计论、功利主义和康德论证的核心结构。熟记它们将为你的分析与评价题提供可靠的工具箱。
1. Modus Ponens (Affirming the Antecedent) | 肯定前件式
→ Q, P, therefore Q. This is the logical engine behind countless philosophical arguments.
肯定前件式是最基本的有效论证形式之一。它表示：如果 P 蕴涵 Q，且 P 为真，那么 Q 必然为真。用符号表示：P → Q，P，所以 Q。这是无数哲学论证背后的逻辑引擎。
For example, if we say 'If the soul is immortal (P), then it survives death (Q)' and we accept that 'the soul is immortal' is true, we must conclude the soul survives death. AQA questions often ask you to identify the logical structure of an argument; modus ponens is frequently at work.
例如，如果我们说“如果灵魂不朽（P），那么它死后继续存在（Q）”，并且我们接受“灵魂不朽”为真，就必须得出结论：灵魂死后继续存在。AQA 考题常要求你识别论证的逻辑结构；肯定前件式经常在其中起作用。
2. Modus Tollens (Denying the Consequent) | 否定后件式
orm. It states: if P implies Q, and Q is false, then P must be false. Symbolically: P → Q, ¬Q, therefore ¬ crucial for arguments that try to disprove a claim by showing its consequences are false.
否定后件式是与之互补的有效形式。它表示：如果 P 蕴涵 Q，而 Q 为假，那么 P 必然为假。符号表示为：P → Q，¬Q，所以 ¬P。这种形式对于那些试图通过表明某个主张的推论为假来反驳它的论证至关重要。
Consider the Problem of Evil: 'If an all‑powerful, all‑loving God exists (P), then there would be no unnecessary suffering (Q).' An atheist might then argue 'But there is unnecessary suffering (¬Q), therefore no such God exists (¬P).' That is a classic modus tollens.
思考一下恶的问题：“如果全知全能且全善的上帝存在（P），那么就不会有无谓的苦难（Q）。”无神论者可能会接着论证“但确实存在无谓的苦难（¬Q），因此这样的上帝不存在（¬P）。”这就是一个典型的否定后件式。
3. Anselm’s Ontological Argument Structure | 安瑟伦本体论论证结构
Anselm’s argument is not a logical formula in the modern sense, but it can be captured in a sequence that functions like a proof. The core moves are: (1) God is defined as 'that than which nothing greater can be conceived'. (2) Existing in reality is greater than existing only in the mind. (3) If God existed only in the mind, we could conceive of something greater – a being that also exists in reality. (4) That would contradict the definition. Therefore, God must exist in reality.
安瑟伦的论证并非现代意义上的逻辑公式，但可以用一个类似证明的步骤序列来表达。核心步骤是：（1）上帝被定义为“可设想的最伟大存在”。（2）存在于现实中比只存在于心灵中更伟大。（3）如果上帝只存在于心灵中，我们就可以设想一个更伟大的存在——一个也存在于现实中的存在。（4）那将与该定义相矛盾。因此，上帝必然存在于现实中。
You can think of it as: Definition D1 exists in mind; D1 is maximal; mind-only existence is less than real existence; so D1 must have real existence. When evaluating, look for issues like whether 'existence is a predicate' or whether the island parody (Gaunilo) undermines this form.
你可以这样想：定义 D1 存在于心中；D1 是最大化的；仅存于心中不如实际存在伟大；因此 D1 必须实际存在。在评价时，要留意“存在是否是谓词”的问题，或者是岛屿类比（高尼罗）是否削弱了这一形式。
4. The Design Argument (Analogical Form) | 设计论证（类比形式）
William Paley’s watchmaker argument uses a formal analogy: (A) A watch has complex parts organised for a purpose, implying a watchmaker. (B) The natural world exhibits even greater complexity and purpose (e.g., the eye). (C) Similar effects suggest similar causes. (D) Therefore, the natural world implies a cosmic designer – God.
威廉·佩利的手表匠论证使用了一种形式类比：（A）手表拥有为某个目的而组织起来的复杂部件，这意味着有一个造表匠。（B）自然界表现出更巨大的复杂性与目的性（例如眼睛）。（C）相似的结果暗示相似的原因。（D）因此，自然界意味着有一位宇宙设计者——上帝。
This can be summarised with an analogical formula: Object X has property P (complex, purpose‑driven) and has cause C (intelligent designer). Object Y (the universe) also has property P. Hence Y probably has cause C. Hume’s criticisms target the weakness of the analogy and the leap to a single divine designer.
这可以用一个类比公式来概括：物体 X 拥有属性 P（复杂、由目的驱动）并存在原因 C（理智设计者）。物体 Y（宇宙）也拥有属性 P。因此 Y 很可能也有原因 C。休谟的批判正是针对这类比的薄弱之处以及向唯一神圣设计者的跳跃。
5. The Problem of Evil Logical Form | 恶的问题逻辑形式
The logical problem of evil claims that the existence of evil is logically incompatible with the existence of the God of classical theism. The formula: (1) If God exists, He is omnipotent, omniscient, and omnibenevolent. (2) An omnipotent being can prevent all evil; an omniscient being knows when evil occurs; an omnibenevolent being would want to prevent all evil. (3) Yet evil exists. (4) Therefore, such a God does not exist.
恶的逻辑问题声称，恶的存在与古典有神论的上帝存在在逻辑上无法相容。公式为：（1）如果上帝存在，他就是全能、全知、全善的。（2）全能者能阻止所有恶；全知者知道何时发生恶；全善者会愿意阻止所有恶。（3）但恶存在。（4）因此，这样的上帝不存在。
This can be structured as a reductio ad absurdum: Assume God (with those attributes) and evil both exist; deduce a contradiction; reject the assumption. AQA expects you to know the free‑will defence: God might allow evil for the greater good of free will, which does not contradict omnibenevolence because free will is a higher‑order good.
这可以构造为归谬法：假设上帝（拥有这些属性）和恶同时存在；推导出矛盾；否定该假设。AQA 希望你知道自由意志辩护：上帝可能为了自由意志这一更大的善而容许恶，这与全善并不矛盾，因为自由意志是一种更高阶的善。
6. Utilitarian Calculus (The Hedonic Calculus) | 功利主义计算（快乐计算）
Jeremy Bentham’s utilitarianism proposes a 'formula' for moral decision‑making: the right action is the one that produces the greatest happiness for the greatest number. His hedonic calculus evaluates pleasures along seven dimensions: intensity, duration, certainty, propinquity (nearness), fecundity (chance of leading to more pleasure), purity (freedom from pain), and extent (number affected).
杰里米·边沁的功利主义提出了一个道德决策的“公式”：正确的行为就是为最大多数人带来最大幸福的行为。他的快乐计算法从七个维度评估快乐：强度、持续时间、确定性、临近度（远近）、丰产性（导致更多快乐的可能性）、纯度（是否免于痛苦）和广度（受影响的人数）。
rnative action, and this net balance benefits the most people. Mill later refined the idea by distinguishing higher and lower pleasures, adding a qualitative element to the calculation.
我们可以将这表示为：如果行动 A 带来的 Σ（快乐 × 各因素）大于任一替代行动，且这个净余额使最多人受益，则行动 A 在道德上是正确的。密尔后来通过区分高级快乐与低级快乐对这个想法加以精炼，为计算加上了质的要素。
7. Kant’s Categorical Imperative Formula | 康德的绝对命令公式
Kant’s moral philosophy is built around the categorical imperative, which has several formulations. The universal law formula states: 'Act only according to that maxim whereby you can at the same time will that it should become a universal law.' This is a test of consistency: if a maxim’s universalisation leads to a contradiction in conception or in the will, the action is morally impermissible.
康德的道德哲学围绕绝对命令展开，它有好几种表述。普遍法则公式是：“只按照你同时能够意愿意愿它成为普遍法则的准则去行动。”这是一个一致性测试：如果一条准则的普遍化导致概念上的矛盾或意志上的矛盾，那么该行动在道德上就是不允许的。
For instance, making a false promise: Can I will that everyone makes false promises? If everyone did, the institution of promising would collapse, so I cannot rationally will it. Thus the maxim fails the test. In exams, you can present this as a two‑step formula: (1) Formulate the maxim, (2) Test its universalisability. If fails, the act is wrong.
例如，作虚假承诺：我能意愿每个人都作虚假承诺吗？如果每个人都这么做，承诺制度就会崩溃，因此我无法理性地意愿它。于是该准则未能通过测试。考试中，你可以将此展示为一个两步公式：（1）制定准则，（2）检验其可普遍化性。若失败，则该行为是错误的。
8. Descartes’ Cogito Argument Form | 笛卡尔“我思故我在”论证形式
Descartes’ famous 'Cogito ergo sum' emerges from the method of radical doubt. The formula: I am thinking (even if I doubt, I am doubting – which is a form of thinking). If I am thinking, there must be a thinker – a self that thinks. Therefore, I exist (at least as a thinking thing). The argument self‑verifies because the very act of doubting one’s existence confirms it.
笛卡尔著名的“我思故我在”源于彻底怀疑的方法。公式为：我正在思想（即便我怀疑，我正在怀疑——它就是思想的一种形式）。如果我在思想，就必然有一个思想者——一个在思想的自我。因此，我存在（至少作为一个思想之物）。这个论证是自我验证的，因为质疑自身存在这一行为本身就确证了它。
This can be symbolised: Let D be the act of doubting. D occurs. D entails T (thinking). T entails existence of a thinking subject S. ∴ S exists. Even an evil demon cannot deceive me about my own existence because deception requires me to exist to be deceived. This foundational certainty is often examined alongside criticisms (e.g., does it prove a substantive self or only momentary awareness?).
这可以用符号表示：令 D 为怀疑行为。D 发生。D 蕴含 T（思想）。T 蕴含思想主体 S 的存在。所以 S 存在。即使有一个邪恶精灵也无法在“我存在”这件事上欺骗我，因为欺骗要求我存在才能被欺骗。这个基础确定性常常与批评一起考查（例如，它证实的是一个恒常的自我抑或只是瞬间的意识？）。
9. Plato’s Theory of Forms Simplified | 柏拉图理型论简化公式
Plato’s theory can be expressed as a formula linking the visible world to an intelligible realm. The relationship is: Particular objects in the visible world (A) participate in (B) the Forms (C), which are perfect, eternal, unchanging archetypes. Knowledge (D) is only possible through rational grasping of the Forms, not through sensory experience. For example, all beautiful things participate in the Form of Beauty itself.
柏拉图的理论可以表述为一个将可见世界与可知领域联系起来的公式。该关系是：可见世界中的个别事物（A）分有（B）理型（C），理型是完美、永恒、不变的范型。知识（D）只有通过对理型的理性把握才可能获得，通过感官经验则不行。例如，一切美的事物都分有“美本身”这一理型。
The 'formula' for a philosophical explanation is: Why are many x’s all x? Because they share in the Form of X. This explanatory structure is applied to beauty, justice, largeness, and so on. Keep in mind Aristotle’s criticism (the third man argument) which questions whether the Form needs a further Form to explain it, leading to infinite regress.
哲学解释的“公式”是：为什么许多 x 都是 x？因为它们都分有 X 的理型。这一解释结构被应用于美、正义、大等概念。要记住亚里士多德的批评（第三人论证），它质疑理型是否又需要一个更高的理型来解释它，从而导向无穷倒退。
10. Pascal’s Wager Decision Matrix | 帕斯卡赌注决策矩阵
Pascal’s Wager presents belief in God as a rational bet with potentially infinite reward. It can be displayed as a decision matrix. The idea is: if you believe in God and God exists, you gain infinite happiness (heaven). If you believe and God does not exist, you lose little. If you do not believe and God exists, you face infinite loss (hell). If you do not believe and God does not exist, you gain little. So believing dominates.
帕斯卡的赌注将信仰上帝展现为一个可能带来无限回报的理性下注。它可以用一个决策矩阵来展示。其思想是：若你信仰上帝且上帝存在，你会获得无限幸福（天堂）；若你信仰而上帝不存在，你几乎无损；若你不信而上帝存在，你将面临无限损失（地狱）；若你不信且上帝不存在，你几乎无得。因此，信仰是占优选择。
Here is the simplified matrix:
| God exists | God does not exist | |
|---|---|---|
| Believe | +∞ (heaven) | Finite loss |
| Disbelieve | -∞ (hell) | Finite gain |
This formalisation makes it easier to evaluate criticisms: the many‑gods objection (which God?), the impossibility of genuine voluntary belief, and the problem of whether God would reward such calculating faith.
这种形式化让评价变得更容易：多神反驳（信哪个神？）、真正自愿信仰的不可能，以及上帝是否会奖赏这种算计性信仰的问题。
Published by TutorHao | Philosophy Revision Series | aleveler.com
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